Statistical Convergence of Nets Through Directed Sets
Yıl 2019,
Cilt: 2 Sayı: 2, 79 - 84, 28.06.2019
Ar Murugan
J. Dianavinnarasi
C. Ganesa Moorthy
Öz
The concept of statistical convergence based on asymptotic density is introduced in this article through nets. Some possible extensions of classical results for statistical convergence of sequences are obtained in this article, with extensions to nets.
Kaynakça
- [1] H. Albayrak, S. Pehlivan, Statistical convergence and statistical continuity on locally solid Riesz spaces, Topol. Appl., 159 (2012), 1887-1893.
- [2] A. Alotaibi, A. M. Alroqi, Statistical convergence in a paranorned space, J. Inequal. Appl., 39 (2012), 6 pages, doi: 10.1186/1029-242X-2012-39 .
- [3] B. Bilalov, T. Nazarova, On statistical convergence in metric spaces, J. Math. Res., 7(1) (2015), 37-43.
- [4] B. Bilalov, T. Nazarova, On statistical type convergence in uniform spaces, Bull. of the Iranian Math. Soc., 42(4) (2016), 975-986.
- [5] R. C. Buck, Generalized asymptotic density, Amer. J. Math., 75 (1953), 335-346.
- [6] H. Cakalli, On statistical convergence in topological groups, Pure Appl. Math. Sci., 43 (1996), 27-31.
- [7] E. D¨undar, Y. Sever, Multipliers for bounded statistical convergence of double sequences, Int. Math. Forum., 7(52) (2012), 2581-2587.
- [8] E. D¨undar, U. Ulusu, B. Aydin, I2-lacunary statistical convergence of double sequences of sets, Konuralp J. Math., 5(1) (2017), 1-10.
- [9] E. D¨undar, U. Ulusu, F. Nuray, On ideal invariant convergence of double sequences and some properties, Creat. Math. Inform., 27(2) (2018), 161-169.
- [10] H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241-244.
- [11] J. A. Fridy, On statistical convergence, Anal., 5 (1985), 301-313.
- [12] J. A. Fridy, Statistical limit points, Proc. Amer. Math. Soc., 118(4) (1993), 1187-1192.
- [13] J. L. Kelly, General topology, Springer, (1975).
- [14] E. Kolk, The statistical convergence in Banach spaces, Acta Comment. Univ. Tartu. Math., 928 (1991), 41-52.
- [15] P. Kostyrko, W. Wilcznski, T. Salat, I-convergence, Real Anal. Exchange, 26(2) (2000), 669-686.
- [16] B. K. Lahiri, P. Das, I and I-convergence in topological spaces, Math. Bohem., 130(2) (2005), 153-160.
- [17] B. K. Lahiri, P. Das, I and I-convergence of nets, Real Anal. Exchange, 33(2) (2007-2008), 431-442.
- [18] S. Loganathan, C. G. Moorthy, A net convergence for Schauder double bases, Asian-Eur. J. Math., 9(1) (2016), 1650010.
- [19] S. Loganathan, C. G. Moorthy, Block convergence of series in topological vector spaces, J. Ana. Num. Theor., 4(1) (2016), 61-69.
- [20] I. J. Maddox, Statistical convergence in a locally convex space, Math. Cambridge Phil. Soc., 104(1) (1988), 141-145.
- [21] G. D. Maio, L. D. R. Kocinac, Statistical convergence in topology, Topol. Appl., 156 (2008), 28-45.
- [22] C. G. Moorthy, A problem of Good on Hausdorff dimension, Mathematika, 39(2) (1992), 244-246.
- [23] C. G. Moorthy, R. Vijaya, P. Venkatachalapathy, Hausdorff dimension of Cantor-like sets, Kyungpook Math. J., 32(2) (1992), 197-202.
- [24] C. G. Moorthy, I. Raj, Weak convergence of fixed point iterations in metric spaces, J. Optimiz. Theory App., 4(2) (2013), 189-192.
- [25] C. G. Moorthy, T. Ramasamy, Pringsheim convergence of double sequences for uniform boundedness principle, Asian-Eur. J. Math., 10(4) (2017),
1750080.
- [26] M. Mursaleen, O. H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl., 288 (2003), 223-231.
- [27] M. Mursaleen, O. H. H. Edely, Generalized statistical convergence, Inform. Sci., 162(3-4) (2004), 287-294.
- [28] F. Nuray, U. Ulusu, E. Dündar, Lacunary statistical convergence of double sequences of sets, Soft Comput., 20(7) (2016), 2883-2888.
- [29] N. Pancarolu, E. Dündar, U. Ulusu, Asymptotically Isq -statistical equivalence of sequences of sets defined by a modulus functions, Sakarya Univ. J. Sci.,
22(6) (2018), 1857-1862.
- [30] D. Rath, B. C. Tripathy, On statistically convergent and statistically Cauchy sequences, Indian J. Pure appl. Math., 25(4) (1994), 381-386.
- [31] T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca, 30(2) (1980), 139-150.
- [32] E. Savas, P. Das, A generalized statistical convergence via ideals, Appl. Math. Lett., 24(6) (2011), 826-830.
- [33] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66(5) (1959), 361-375.
- [34] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2 (1951), 73-74.
- [35] U. Ulusu, F. Nuray, Lacunary statistical convergence of sequences of sets, Prog. Appl. Math., 4(2) (2012), 99-109.
- [36] U. Ulusu, E. Dündar, I-Lacunary statistical convergence of sequences of sets, Filomat, 28(8) (2014), 1567-1574.
- [37] U. Ulusu, Asymptotically ideal invariant equivalence, Creat. Math. Inform., 27(2) (2018), 215-220.
- [38] S. Yegul, E. Dündar, On statistical convergence of sequences of functions in 2-normed spaces, J. Class. Anal., 10(1) (2017), 49-57.
- [39] S. Yegul, E. Dündar, Statistical Convergence of Double Sequences of Functions and Some Properties In 2-Normed Spaces, Facta Universitatis, Ser.
Math. and Infor., 33(5) (2018), 705-719.
Yıl 2019,
Cilt: 2 Sayı: 2, 79 - 84, 28.06.2019
Ar Murugan
J. Dianavinnarasi
C. Ganesa Moorthy
Kaynakça
- [1] H. Albayrak, S. Pehlivan, Statistical convergence and statistical continuity on locally solid Riesz spaces, Topol. Appl., 159 (2012), 1887-1893.
- [2] A. Alotaibi, A. M. Alroqi, Statistical convergence in a paranorned space, J. Inequal. Appl., 39 (2012), 6 pages, doi: 10.1186/1029-242X-2012-39 .
- [3] B. Bilalov, T. Nazarova, On statistical convergence in metric spaces, J. Math. Res., 7(1) (2015), 37-43.
- [4] B. Bilalov, T. Nazarova, On statistical type convergence in uniform spaces, Bull. of the Iranian Math. Soc., 42(4) (2016), 975-986.
- [5] R. C. Buck, Generalized asymptotic density, Amer. J. Math., 75 (1953), 335-346.
- [6] H. Cakalli, On statistical convergence in topological groups, Pure Appl. Math. Sci., 43 (1996), 27-31.
- [7] E. D¨undar, Y. Sever, Multipliers for bounded statistical convergence of double sequences, Int. Math. Forum., 7(52) (2012), 2581-2587.
- [8] E. D¨undar, U. Ulusu, B. Aydin, I2-lacunary statistical convergence of double sequences of sets, Konuralp J. Math., 5(1) (2017), 1-10.
- [9] E. D¨undar, U. Ulusu, F. Nuray, On ideal invariant convergence of double sequences and some properties, Creat. Math. Inform., 27(2) (2018), 161-169.
- [10] H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241-244.
- [11] J. A. Fridy, On statistical convergence, Anal., 5 (1985), 301-313.
- [12] J. A. Fridy, Statistical limit points, Proc. Amer. Math. Soc., 118(4) (1993), 1187-1192.
- [13] J. L. Kelly, General topology, Springer, (1975).
- [14] E. Kolk, The statistical convergence in Banach spaces, Acta Comment. Univ. Tartu. Math., 928 (1991), 41-52.
- [15] P. Kostyrko, W. Wilcznski, T. Salat, I-convergence, Real Anal. Exchange, 26(2) (2000), 669-686.
- [16] B. K. Lahiri, P. Das, I and I-convergence in topological spaces, Math. Bohem., 130(2) (2005), 153-160.
- [17] B. K. Lahiri, P. Das, I and I-convergence of nets, Real Anal. Exchange, 33(2) (2007-2008), 431-442.
- [18] S. Loganathan, C. G. Moorthy, A net convergence for Schauder double bases, Asian-Eur. J. Math., 9(1) (2016), 1650010.
- [19] S. Loganathan, C. G. Moorthy, Block convergence of series in topological vector spaces, J. Ana. Num. Theor., 4(1) (2016), 61-69.
- [20] I. J. Maddox, Statistical convergence in a locally convex space, Math. Cambridge Phil. Soc., 104(1) (1988), 141-145.
- [21] G. D. Maio, L. D. R. Kocinac, Statistical convergence in topology, Topol. Appl., 156 (2008), 28-45.
- [22] C. G. Moorthy, A problem of Good on Hausdorff dimension, Mathematika, 39(2) (1992), 244-246.
- [23] C. G. Moorthy, R. Vijaya, P. Venkatachalapathy, Hausdorff dimension of Cantor-like sets, Kyungpook Math. J., 32(2) (1992), 197-202.
- [24] C. G. Moorthy, I. Raj, Weak convergence of fixed point iterations in metric spaces, J. Optimiz. Theory App., 4(2) (2013), 189-192.
- [25] C. G. Moorthy, T. Ramasamy, Pringsheim convergence of double sequences for uniform boundedness principle, Asian-Eur. J. Math., 10(4) (2017),
1750080.
- [26] M. Mursaleen, O. H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl., 288 (2003), 223-231.
- [27] M. Mursaleen, O. H. H. Edely, Generalized statistical convergence, Inform. Sci., 162(3-4) (2004), 287-294.
- [28] F. Nuray, U. Ulusu, E. Dündar, Lacunary statistical convergence of double sequences of sets, Soft Comput., 20(7) (2016), 2883-2888.
- [29] N. Pancarolu, E. Dündar, U. Ulusu, Asymptotically Isq -statistical equivalence of sequences of sets defined by a modulus functions, Sakarya Univ. J. Sci.,
22(6) (2018), 1857-1862.
- [30] D. Rath, B. C. Tripathy, On statistically convergent and statistically Cauchy sequences, Indian J. Pure appl. Math., 25(4) (1994), 381-386.
- [31] T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca, 30(2) (1980), 139-150.
- [32] E. Savas, P. Das, A generalized statistical convergence via ideals, Appl. Math. Lett., 24(6) (2011), 826-830.
- [33] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66(5) (1959), 361-375.
- [34] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2 (1951), 73-74.
- [35] U. Ulusu, F. Nuray, Lacunary statistical convergence of sequences of sets, Prog. Appl. Math., 4(2) (2012), 99-109.
- [36] U. Ulusu, E. Dündar, I-Lacunary statistical convergence of sequences of sets, Filomat, 28(8) (2014), 1567-1574.
- [37] U. Ulusu, Asymptotically ideal invariant equivalence, Creat. Math. Inform., 27(2) (2018), 215-220.
- [38] S. Yegul, E. Dündar, On statistical convergence of sequences of functions in 2-normed spaces, J. Class. Anal., 10(1) (2017), 49-57.
- [39] S. Yegul, E. Dündar, Statistical Convergence of Double Sequences of Functions and Some Properties In 2-Normed Spaces, Facta Universitatis, Ser.
Math. and Infor., 33(5) (2018), 705-719.